Bet bigger on the flop, $6 (or even potsize) should be fine. You want to protect your hand and get value before your opponent gets scared with his low pp when a broadway card hits the turn (as it did).

Originally posted by Kaitz20
check behind on the turn is fine, but then you should call river bet.

Sorry but that doesn't make sense at all, you should definitely be betting the turn.
Still for value, protection, because you will call most rivers anyway and because you maybe want to 2nd barrel this kind of card when you have air.

And btw calling a bet on this river is suicidal, there is no reason for the opponent to turn a middle pocket pair into a bluff, you'll be beaten the very most of the time here.